Abstract

We investigate a 𝐻1-Galerkin mixed finite element method for nonlinear viscoelasticity equations based on 𝐻1-Galerkin method and expanded mixed element method. The existence and uniqueness of solutions to the numerical scheme are proved. A priori error estimation is derived for the unknown function, the gradient function, and the flux.

1. Introduction

Consider the following nonlinear viscoelasticity-type equation:
𝑢𝑡𝑡𝑎−∇⋅(𝑥,𝑢)∇𝑢𝑡+𝑏(𝑥,𝑢)∇𝑢=𝑓(𝑥,𝑡),(𝑥,𝑡)∈Ω×𝐽,𝑢(𝑥,𝑡)=0,(𝑥,𝑡)∈𝜕Ω×𝐽,𝑢(𝑥,0)=𝑢0(𝑢𝑥),𝑥∈Ω,𝑡(𝑥,0)=𝑢1(𝑥),𝑥∈Ω,(1.1)
where Ω is a convex polygonal domain in 𝑅2 with the Lipschitz continuous boundary 𝜕Ω, 𝐽=(0,𝑇] is the time interval with 0<𝑇<∞, and 𝑢0(𝑥) and 𝑢1(𝑥) are, respectively, the initial data functions defined on Ω. The deformation of viscoelastic solid under the external loads is usually considered by means of this viscoelastic model [1–4], and the problem has a unique sufficiently smooth solution with the regularity condition provided that the given data 𝑢0(𝑥), 𝑢1(𝑥), 𝑎(𝑢), 𝑏(𝑢), and 𝑓 are sufficiently smooth [5].

For problem (1.1), by adopting finite element method, Lin et al. [6] established the convergence of the finite element approximations to solutions of Sobolev and viscoelasticity type of equations via Ritz-Volterra projection and an optimal-order error estimates in 𝐿𝑝 (2≤𝑝<∞). Latter, Lin and Zhang [7] presented a direct analysis for global superconvergence for this problem without using the Ritz projection or its modified forms. Jin et al. [8] and Shi et al. [9] employed the Wilson nonconforming finite element and a Crouzeix-Raviart type nonconforming finite element on the anisotropic meshes to solve viscoelasticity-type equations, and the global superconvergence estimations were obtained by means of post-processing technique. Since the estimation of flux ∇𝑢 by the unknown scalar 𝑢 is usually indirect, thus the quantity of calculation of the finite element method is relatively large.

As an efficient strategy, mixed finite element methods received much attention in solving partial differential equation in recent decades [10–16]. Compared with finite element methods, mixed finite element methods can obtain the unknown scalar 𝑢 and its flux ∇𝑢 directly, and; hence, it can decrease smoothness of solution space. However, the LBB assumption is needed in the approximating subspaces and; hence, confines the choice of finite element spaces.

On the base of the mixed finite element methods, Pani [17] proposed a new mixed finite element method, called the 𝐻1-Galerkin mixed finite element procedure, to solve a mixed system in unknown scalar and its flux. Compared with the standard mixed finite methods, the new mixed finite element method does not require the LBB condition, and a better order of convergence for the flux in 𝐿2 norm can be obtained if an extra regularity on the solution holds. Recently, 𝐻1-Galerkin mixed finite element methods were applied to differential equations [18–22]. However, the assumption needed for this method is not suitable for the nonlinear equations and equations with a small tensor. To overcome this, Chen and Wang [23] proposed 𝐻1-Galerkin expanded mixed finite element methods which combines the 𝐻1-Galerkin formulation and the expanded mixed finite element methods [24] to deal with a nonlinear parabolic equation in porous medium flow. This method can compute the scalar unknown, its gradient, and its flux directly. Hence, it is suitable to the case where the coefficient of the differential equation is a small tensor and cannot be inverted. Motivated by this, we establish an 𝐻1-Galerkin expanded mixed finite element method for the viscoelasticity-type equations.

The remainder of this paper is organized as follows. In Section 2, we first establish the equivalence between viscoelasticity-type equations and their weak formulation by using the 𝐻1-Galerkin expanded mixed finite element methods and then discuss the existence and uniqueness of the formulation. In Section 3, we show that the 𝐻1-Galerkin expanded mixed finite element method has the same convergence rate as that of the classical mixed finite element methods without requiring the LBB consistency condition.

Throughout this paper, we use 𝐻 to denote the space 𝐻(div,Ω)={𝐯∈(𝐿2(Ω))𝑑∶∇⋅𝐯∈𝐿2(Ω)} with norm ‖𝐯‖𝐇(div;Ω)=(‖𝐯‖2+‖∇⋅𝐯‖2)1/2 and 𝐻10(Ω)={𝑤∈𝐻1(Ω)∶𝑤=0on𝜕Ω}. For theoretical analysis, we also need the following assumptions on the functions involved in problem (1.1).

Assumption 1.1. (1) There exist constants a1 and a2 such that 0<𝑎1≤𝑎(𝑥,𝑢),𝑏(𝑥,𝑢)≤𝑎2. (2) The functions 𝑎(𝑥,𝑢),𝑏(𝑥,𝑢),𝑎𝑢(𝑥,𝑢), and 𝑏𝑢(𝑥,𝑢) are Lipschitz continuous with respect to 𝑢, and there exists 𝐶1>0 such that |𝜕𝑎/𝜕𝑢|+|𝜕𝑏/𝜕𝑢|+|𝜕2𝑎/𝜕𝑢2|+|𝜕2𝑏/𝜕𝑢2|≤𝐶1.

2. 𝐻1-Galerkin Expanded Mixed Finite Element Discrete Scheme

2.1. Weak Formulation

To define the 𝐻1-Galerkin expanded mixed finite element procedure, we introduce vector
𝐩=𝑎(𝑥,𝑢)∇𝑢𝑡+𝑏(𝑥,𝑢)∇𝑢,𝝈=∇𝑢,(2.1)
and split (1.1) into a first-order system as follows:
𝑢𝑡𝑡−∇⋅𝐩=𝑓,𝝈=∇𝑢,𝐩=𝑎(𝑢)𝝈𝑡+𝑏(𝑢)𝝈,𝝈(𝑥,0)=∇𝑢0𝝈(𝑥),𝑡(𝑥,0)=∇𝑢1𝑢(𝑥),𝐩(𝑥,0)=𝑎0∇𝑢1𝑢(𝑥)+𝑏0∇𝑢0(𝑥).(2.2)
Then by Green's formula we can further define the following weak formulation of problem (2.2): find (𝑢,𝝈,𝐩)∈𝐻10(Ω)×𝐻(div,Ω)×𝐻(div,Ω) such that
𝝈𝑡𝑡+,𝐪(∇⋅𝐩,∇⋅𝐪)=−(𝑓,∇⋅𝐪),∀𝐪∈𝐻(div,Ω),(𝝈,∇𝑣)=(∇𝑢,∇𝑣),∀𝑣∈𝐻10(Ω),(𝐩,𝐰)=𝑎(𝑢)𝝈𝑡,𝐰+(𝑏(𝑢)𝝈,𝐰),∀𝐰∈𝐻(div,Ω),𝝈(𝑥,0)=∇𝑢0𝝈(𝑥),𝑡(𝑥,0)=∇𝑢1𝑢(𝑥),𝐩(𝑥,0)=𝑎0∇𝑢1(𝑢𝑥)+𝑏0∇𝑢0(𝑥).(2.3)

In order to establish the equivalence between problem (2.2) and the weak form (2.3), we need the following technical lemmas.

Lemma 2.1 (see [25]). Let Ω be a bounded domain with a Lipschitz continuous boundary 𝜕Ω. Then, for any 𝐩∈𝐻(div,Ω), there exists 𝜙∈𝐻2⋂𝐻(Ω)10(Ω) and divergence free 𝜓∈𝐻(div,Ω) such that ∇⋅𝜓=0 and 𝐩=∇𝜙+𝜓.

Lemma 2.2 (see [26]). Let Ω be a bounded domain with a Lipschitz continuous boundary 𝜕Ω. Then, for any 𝑔∈𝐿2(Ω), there exists 𝐩∈(𝐻1(Ω))𝑑⊂𝐻(div,Ω) such that ∇⋅𝐩=𝑔.

Now we are in a position to state our main result in this subsection.

Theorem 2.3. Under the conditions of Lemmas 2.1 and 2.2, (𝑢,𝝈,𝐩)∈𝐻10(Ω)×𝐻(div,Ω)×𝐻(div,Ω) is a solution to the system (2.2) if and only if it is a solution to the weak form (2.3).

Proof. It is easy to check that any solution to the system (2.2) is a solution to the weak form (2.3). Hence, to prove the assertion, we only need to show that any solution to the weak form (2.3) is a solution to the system (2.2).First, taking 𝐰=𝐩−𝑎(𝑢)𝝈𝑡−𝑏(𝑢)𝝈 in the third equation of (2.3) leads to
𝐩−𝑎(𝑢)𝝈𝑡−𝑏(𝑢)𝝈,𝐩−𝑎(𝑢)𝝈𝑡−𝑏(𝑢)𝝈=0,(2.4)
which implies
𝐩=𝑎(𝑢)𝝈𝑡−𝑏(𝑢)𝝈.(2.5)By Lemma 2.1, there exist 𝜙∈𝐻2⋂𝐻(Ω)10(Ω) and divergence free 𝜓∈𝐻(div,Ω) such that ∇⋅𝜓=0 and 𝝈=∇𝜙+𝜓. Choosing 𝝈=∇𝜙+𝜓 in the second equation of (2.3) yields
(∇𝜙+𝜓,∇𝑣)=(∇𝑢,∇𝑣),∀𝑣∈𝐻10(Ω).(2.6)
By the divergence theorem [1], one has
(𝜓,∇𝑣)=−(∇⋅𝜓,𝑣)=0,∀𝑣∈𝐻10(Ω).(2.7)
Substituting (2.7) into (2.6) yields
(∇𝜙,∇𝑣)=(∇𝑢,∇𝑣),∀𝑣∈𝐻10(Ω),(2.8)
which means that
∇𝜙=∇𝑢,𝝈=∇𝑢+𝜓.(2.9)
Inserting (2.5) and (2.9) into the first equation of (2.2) and applying the divergence theorem to the first term, for any 𝐪∈𝐻(div,Ω), one has
𝑢𝑡𝑡−𝜓,∇⋅𝐪𝑡𝑡−𝑎,𝐪∇⋅(𝑢)∇𝑢𝑡+𝜓𝑡+,∇⋅𝐪(∇⋅(𝑏(𝑢)(∇𝑢+𝜓)),∇⋅𝐪)=(𝑓,∇⋅𝐪).(2.10)
Instituting 𝐪=𝜓𝑡 into (2.10) and using ∇⋅𝜓𝑡=0 lead to
𝜓0=𝑡𝑡,𝜓𝑡=12𝑑𝜓𝑑𝑡𝑡,𝜓𝑡.(2.11)
Integrating from 0 to 𝑡 with respect to time results in
𝜓𝑡(𝑥,𝑡),𝜓𝑡=𝜓(𝑥,𝑡)𝑡(𝑥,0),𝜓𝑡.(𝑥,0)(2.12)
Differentiating (2.9) with respect to 𝑡, one obtains
𝝈𝑡=∇𝑢𝑡+𝜓𝑡.(2.13)
By the fifth equation in (2.3), we deduce that
𝜓𝑡,(𝑥,0)=0(2.14)
which implies
𝜓𝑡(𝑥,𝑡)=0.(2.15)
Integrating the equation 𝜓𝑡(𝑥,𝑡)=0 with respect to 𝑡 from 0 to 𝑡 gives
𝜓(𝑥,𝑡)=𝜓(𝑥,0).(2.16)
By (2.9) and the forth equation in (2.2), we deduce
𝜓(𝑥,𝑡)=0,(2.17)
which leads to
𝝈=∇𝑢.(2.18)
Therefore, (2.10) can equivalently be transformed into the following equation:
𝑢𝑡𝑡−𝑎,∇⋅𝐪∇⋅(𝑢)∇𝑢𝑡=+𝑏(𝑢)∇𝑢,∇⋅𝐪(𝑓,∇⋅𝐪),∀𝐪∈𝐻(div,Ω).(2.19)
For 𝑓,𝑢𝑡𝑡∈𝐿2(Ω), by Lemma 2.2, there exists 𝐅∈𝐻(div,Ω) such that ∇⋅𝐅=𝑢𝑡𝑡−𝑓. Thus, (2.19) reduces to
(∇⋅𝐩,∇⋅𝐪)=(∇⋅𝐅,∇⋅𝐪),∀𝐪∈𝐻(div,Ω).(2.20)
Recalling Lemma 2.1, one concludes that
∇⋅𝐅=∇⋅𝐩,(2.21)
that is,
𝑢𝑡𝑡−∇⋅𝐩=𝑓.(2.22)
Combining this with (2.5) and (2.18) results in the desired assertion, and this completes the proof.

2.2. Numerical Scheme

Let 𝑇ℎ be a quasi-uniform family of subdivision of domain Ω; that is, Ω=∪𝐾∈𝑇ℎ𝐾 with ℎ = max {diam(𝐾)∶𝐾∈𝑇ℎ}, and let 𝑉ℎ be the finite-dimensional subspaces of 𝐻10(Ω) defined by 𝑉ℎ=𝑣ℎ∈𝐻10(Ω);𝑣ℎ∣𝐾∈𝑃𝑚,(𝐾)(2.23)
where 𝑃𝑚(𝐾) denotes the space of polynomials of degree at most 𝑚 on 𝐾. Moreover, we denote the vector space in mixed finite element spaces with index 𝑘 by 𝐻ℎ. It is well known that both 𝐻ℎ and 𝑉ℎ satisfy the inverse property and the following approximation properties [26, 27]: inf𝑣ℎ∈𝑉ℎ‖‖𝑣−𝑣ℎ‖‖‖‖+ℎ𝑣−𝑣ℎ‖‖1≤𝐶ℎ𝑚+1‖𝑣‖𝑚+1,𝑣∈𝐻𝑚+1(Ω),inf𝐪ℎ∈𝑊ℎ‖‖𝐪−𝐪ℎ‖‖≤𝐶ℎ𝑘+1‖𝐪‖𝑘+1,𝐪ℎ∈𝐻𝑘+1(Ω)𝑑.(2.24)

With the above notations, the semidiscrete 𝐻1-Galerkin expanded mixed finite element method for system (2.3) is reduced to find a triple (𝑢ℎ,𝝈ℎ,𝐩ℎ)∈𝑉ℎ×𝐻ℎ×𝐻ℎ such that
𝝈ℎ𝑡𝑡,𝐪ℎ+∇⋅𝐩ℎ,∇⋅𝐪ℎ=−𝑓,∇⋅𝐪ℎ,∀𝐪ℎ∈𝐻ℎ,𝝈ℎ,∇𝑣ℎ=∇𝑢ℎ,∇𝑣ℎ,∀𝑣ℎ∈𝑉ℎ,𝐩ℎ,𝐰ℎ=𝑎𝑢ℎ𝝈ℎ𝑡,𝐰ℎ+𝑏𝑢ℎ𝝈ℎ,𝐰ℎ,∀𝐰ℎ∈𝐻ℎ,𝐩ℎ(𝑥,0)=Πℎ𝝈𝐩(𝑥,0),ℎ(𝑥,0)=Πℎ∇𝑢0(𝝈𝑥),ℎ𝑡(𝑥,0)=Πℎ∇𝑢1(𝑥).(2.28)

For the 𝐻1-Galerkin expanded mixed finite element scheme (2.28), we claim that there exists a unique solution.

In fact, set 𝑉ℎ=span{𝜑𝑖}𝑁𝑖=1 and 𝐻ℎ=span{𝜓𝑗}𝑀𝑗=1. Then 𝝈ℎ,𝐩ℎ∈𝐻ℎ and 𝑢ℎ∈𝑉ℎ, and; hence,
𝝈ℎ=𝑀𝑗=1𝑝𝑖(𝑡)𝜓𝑖(𝑥),𝐩ℎ=𝑀𝑗=1𝜆𝑖(𝑡)𝜓𝑖(𝑥),𝑢ℎ=𝑁𝑖=1𝑢𝑖(𝑡)𝜑𝑖(𝑥).(2.29)
Taking 𝐪ℎ=𝜓𝑗, 𝐰ℎ=𝜓𝑗, 𝑗=1,2,…,𝑀, 𝑣ℎ=𝜑𝑖, 𝑖=1,2,…,𝑁 in (2.28) leads to
𝐴𝑃𝑡𝑡+𝐵Λ=𝐹,𝐷𝑈=𝐶𝑃,𝐴Λ=𝑀(𝑈)𝑃𝑡+𝑁(𝑈)𝑃,(2.30)
where𝜓𝐴=𝑖(𝑥),𝜓𝑗(𝑥)𝑀×𝑀𝑝,𝑃=1,𝑝2,…,𝑝𝑀𝑇,𝐵=∇⋅𝜓𝑖(𝑥),∇⋅𝜓𝑗(𝑥)𝑀×𝑀𝜆,Λ=1,𝜆2,…,𝜆𝑀𝑇,𝐷=∇𝜑𝑖(𝑥),∇𝜑𝑗(𝑥)𝑁×𝑁𝑢,𝑈=1,𝑢2,…,𝑢𝑁𝑇,𝜓𝐶=𝑖(𝑥),∇𝜑𝑗(𝑥)𝑁×𝑀𝑎,𝑀(𝑈)=(𝑈)𝜓𝑖(𝑥),𝜓𝑗(𝑥)𝑀×𝑀,𝑁𝑏(𝑈)=(𝑈)𝜓𝑖(𝑥),𝜓𝑗(𝑥)𝑀×𝑀,𝐹=−𝑓,∇⋅𝜓𝑗(𝑥)𝑀×1,(2.31)
and 𝑃(0), 𝑃𝑡(0) are given.

Note that matrix 𝐴 in (2.31) is positive definite. Thus, by the third equation in (2.30), one has
Λ=𝐴−1𝑀𝑃𝑡+𝑁𝑃.(2.32)
Inserting the above equality into the first equation of (2.30) yields
𝐴𝑃𝑡𝑡+𝐵𝐴−1𝑀𝑃𝑡+𝐵𝐴−1𝑁𝑃=𝐹.(2.33)
By the standard arguments on the initial-value problem of a system of ordinary differential equations, we can obtain existence and uniqueness of 𝑃. The existence and uniqueness of 𝑈 and Λ follow from the existence and uniqueness of 𝑃.

3. Error Analysis

This section is devoted to the error estimates for the 𝐻1-Galerkin expanded mixed finite element method.

For error analysis in the following, we need to introduce a projection operator. Let 𝑅ℎ∶𝐻10(Ω)→𝑉ℎ be the Ritz projection defined by
∇𝑢−𝑅ℎ𝑢,∇𝑣ℎ=0,∀𝑣ℎ∈𝑉ℎ.(3.1)
Then the following approximation holds [27]:
‖‖𝑢−𝑅ℎ𝑢‖‖‖‖∇+ℎ𝑢−𝑅ℎ𝑢‖‖≤𝐶ℎ𝑚+1‖𝑢‖𝑚+1.(3.2)

Theorem 3.1. Let (𝑢,𝝈,𝐩) and (𝑢ℎ,𝝈ℎ,𝐩ℎ) be the solutions to (2.3) and (2.28), respectively. Then the following error estimates hold:
‖‖(𝑎)𝑢−𝑢ℎ‖‖1≤𝐶ℎmin(𝑘+1,𝑚),‖‖(𝑏)∇⋅𝝈−𝝈ℎ‖‖≤𝐶ℎmin(𝑘,𝑚+1),‖‖(𝑐)𝑢−𝑢ℎ‖‖+‖‖𝝈−𝝈ℎ‖‖+‖‖𝐩−𝐩ℎ‖‖≤𝐶ℎmin(𝑘+1,𝑚+1),(3.7)
where 𝑘≥1 and 𝑚≥1 for 𝑑=2,3, and the positive constant 𝐶 depends on ‖𝑢𝑡‖𝐿∞(𝐻𝑚+1), ‖𝑢‖𝐿∞(𝐻𝑚+1), ‖𝐩𝑡‖𝐿∞(𝐻𝑘+1), ‖𝐩‖𝐿∞(𝐻𝑘+1), ‖𝝈𝑡‖𝐿∞(𝐻𝑘+1), ‖𝝈𝑡𝑡‖𝐿∞(𝐻𝑘+1), ‖𝝈‖𝐿∞(𝐻𝑘+1).

Proof. Since estimates of 𝜃, 𝜂, and 𝛼 can be obtained by (3.2) and (2.26), it suffices to estimate 𝜉, 𝜁, and 𝛽.Instituting 𝐰ℎ=𝜉𝑡𝑡 into (3.6) and 𝐪ℎ=𝜁 in (3.4) gives
𝑎𝑢(∇⋅𝜁,∇⋅𝜁)+ℎ𝜉𝑡,𝜉𝑡𝑡+𝑏𝑢ℎ𝜉,𝜉𝑡𝑡𝝈=−𝑡𝑎𝑢(𝑢)−𝑎ℎ,𝜉𝑡𝑡−𝝈𝑏𝑢(𝑢)−𝑏ℎ,𝜉𝑡𝑡−𝑎𝑢ℎ𝜃𝑡,𝜉𝑡𝑡−𝑏𝑢ℎ𝜃,𝜉𝑡𝑡+𝜂,𝜉𝑡𝑡−𝜃𝑡𝑡.,𝜁(3.8)
It is easy to check that
𝑎𝑢ℎ𝜉𝑡𝑡,𝜉𝑡=12𝑑𝑎𝑢𝑑𝑡ℎ𝜉𝑡,𝜉𝑡−12𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜉𝑡,𝜉𝑡,𝑏𝑢ℎ𝜉,𝜉𝑡𝑡=𝑑𝑏𝑢𝑑𝑡ℎ𝜉,𝜉𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜉,𝜉𝑡−𝑏𝑢ℎ𝜉𝑡,𝜉𝑡,𝜂,𝜉𝑡𝑡=𝑑𝑑𝑡𝜂,𝜉𝑡−𝜂𝑡,𝜉𝑡,𝝈𝑡𝑢𝑎(𝑢)−𝑎ℎ,𝜉𝑡𝑡=𝑑𝝈𝑑𝑡𝑡𝑢𝑎(𝑢)−𝑎ℎ,𝜉𝑡−𝝈𝑡𝑡𝑢𝑎(𝑢)−𝑎ℎ,𝜉𝑡−𝝈𝑡𝑎𝑢(𝑢)𝑢𝑡−𝑎𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡,𝝈𝑢𝑏(𝑢)−𝑏ℎ,𝜉𝑡𝑡=𝑑𝝈𝑢𝑑𝑡𝑏(𝑢)−𝑏ℎ,𝜉𝑡−𝝈𝑡𝑢𝑏(𝑢)−𝑏ℎ,𝜉𝑡−𝝈𝑏𝑢(𝑢)𝑢𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡,𝑎𝑢ℎ𝜃𝑡,𝜉𝑡𝑡=𝑑𝑎𝑢𝑑𝑡ℎ𝜃𝑡,𝜉𝑡−𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜃𝑡,𝜉𝑡−𝑎𝑢ℎ𝜃𝑡𝑡,𝜉𝑡,𝑏𝑢ℎ𝜃,𝜉𝑡𝑡=𝑑𝑏𝑢𝑑𝑡ℎ𝜃,𝜉𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜃,𝜉𝑡−𝑏𝑢ℎ𝜃𝑡,𝜉𝑡.(3.9)
Thus, (3.8) can be written as
1(∇⋅𝜁,∇⋅𝜁)+2𝑑𝑎𝑢𝑑𝑡ℎ𝜉𝑡,𝜉𝑡+𝑑𝑏𝑢𝑑𝑡ℎ𝜉,𝜉𝑡=12𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜉𝑡,𝜉𝑡+𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜉,𝜉𝑡+𝑏𝑢ℎ𝜉𝑡,𝜉𝑡−𝜃𝑡𝑡+𝑑,𝜁𝑑𝑡𝜂,𝜉𝑡−𝜂𝑡,𝜉𝑡−𝑑𝝈𝑑𝑡𝑡𝑢𝑎(𝑢)−𝑎ℎ,𝜉𝑡+𝝈𝑡𝑡𝑢𝑎(𝑢)−𝑎ℎ,𝜉𝑡+𝝈𝑡𝑎𝑢(𝑢)𝑢𝑡−𝑎𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡−𝑑𝝈𝑢𝑑𝑡𝑏(𝑢)−𝑏ℎ,𝜉𝑡+𝝈𝑡𝑢𝑏(𝑢)−𝑏ℎ,𝜉𝑡+𝝈𝑏𝑢(𝑢)𝑢𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡−𝑑𝑎𝑢𝑑𝑡ℎ𝜃𝑡,𝜉𝑡+𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜃𝑡,𝜉𝑡+𝑎𝑢ℎ𝜃𝑡𝑡,𝜉𝑡−𝑑𝑏𝑢𝑑𝑡ℎ𝜃,𝜉𝑡+𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜃,𝜉𝑡+𝑏𝑢ℎ𝜃𝑡,𝜉𝑡.(3.10)
Integrating this system from 0 to 𝑡 yields
𝑡0‖∇⋅𝜁‖21𝑑𝜏+2𝑎𝑢ℎ𝜉𝑡,𝜉𝑡+𝑏𝑢ℎ𝜉,𝜉𝑡=𝜂,𝜉𝑡−𝝈𝑡𝑢𝑎(𝑢)−𝑎ℎ,𝜉𝑡−𝝈𝑢𝑏(𝑢)−𝑏ℎ,𝜉𝑡−𝑎𝑢ℎ𝜃𝑡,𝜉𝑡−𝑏𝑢ℎ𝜃,𝜉𝑡+12𝑡0𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜉𝑡,𝜉𝑡𝑑𝜏+𝑡0𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜉,𝜉𝑡𝑑𝜏+𝑡0𝑏𝑢ℎ𝜉𝑡,𝜉𝑡−𝑑𝜏𝑡0𝜃𝑡𝑡,𝜁𝑑𝜏−𝑡0𝜂𝑡,𝜉𝑡𝑑𝜏+𝑡0𝝈𝑡𝑡𝑢𝑎(𝑢)−𝑎ℎ,𝜉𝑡+𝑑𝜏𝑡0𝝈𝑡𝑎𝑢(𝑢)𝑢𝑡−𝑎𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡𝑑𝜏+𝑡0𝝈𝑡𝑏𝑢(𝑢)−𝑏ℎ,𝜉𝑡+𝑑𝜏𝑡0𝝈𝑏𝑢(𝑢)𝑢𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡𝑑𝜏+𝑡0𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜃𝑡,𝜉𝑡+𝑑𝜏𝑡0𝑎𝑢ℎ𝜃𝑡𝑡,𝜉𝑡𝑑𝜏+𝑡0𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜃,𝜉𝑡𝑑𝜏+𝑡0𝑏𝑢ℎ𝜃𝑡,𝜉𝑡𝑑𝜏.(3.11)
In what follows, we, respectively, analyze the terms on the right-hand side of (3.11). By the Cauchy-Schwartz inequality, we can bound the sixth term on the right-hand side of (3.11) as follows:
||||𝑡012𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜉𝑡,𝜉𝑡||||=1𝑑𝜏2||||𝑡0𝑎𝑢𝑢ℎ𝑢𝑡𝜉𝑡,𝜉𝑡𝑑𝜏+𝑡012𝑎𝑢𝑢ℎ𝑢ℎ𝑡−𝑢𝑡𝜉𝑡,𝜉𝑡||||𝑑𝜏≤𝐶𝑡0‖‖𝜉𝑡‖‖2‖‖𝜉𝑑𝜏+𝐶𝑡‖‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝛼𝑡‖‖2+‖‖𝛽𝑡‖‖2+‖‖𝜉𝑡‖‖2𝑑𝜏.(3.12)
For the seventh term on the right-hand side of (3.11), one has
||||𝑡0𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜉,𝜉𝑡||||=||||𝑑𝜏𝑡0𝑏𝑢𝑢ℎ𝑢ℎ𝑡−𝑢𝑡𝜉,𝜉𝑡+𝑏𝑢𝑢ℎ𝑢𝑡𝜉,𝜉𝑡||||𝑑𝜏≤𝐶𝑡0‖‖𝜉𝑡‖‖2+‖𝜉‖2‖‖𝜉𝑑𝜏+𝐶𝑡‖‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝛼𝑡‖‖2+‖‖𝛽𝑡‖‖2+‖𝜉‖2𝑑𝜏.(3.13)
For the term ∫𝑡0(𝝈𝑡(𝑎𝑢(𝑢)𝑢𝑡−𝑎𝑢(𝑢ℎ)𝑢ℎ𝑡),𝜉𝑡)𝑑𝜏 on the right side of (3.11), we have
||||𝑡0𝝈𝑡𝑎𝑢(𝑢)𝑢𝑡−𝑎𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡||||=||||𝑑𝜏𝑡0𝝈𝑡𝑎𝑢(𝑢)−𝑎𝑢𝑢ℎ𝑢𝑡+𝑎𝑢𝑢ℎ𝑢𝑡−𝑢ℎ𝑡,𝜉𝑡||||𝑑𝜏≤𝐶𝑡0‖𝛼‖2+‖𝛽‖2+‖‖𝛼𝑡‖‖2+‖‖𝛽𝑡‖‖2+‖‖𝜉𝑡‖‖2𝑑𝜏.(3.14)
Similarly,
||||𝑡0𝝈𝑏𝑢(𝑢)𝑢𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡||||≤𝑑𝜏𝑡0||𝝈𝑏𝑢(𝑢)−𝑏𝑢𝑢ℎ𝑢𝑡+𝑏𝑢𝑢ℎ𝑢𝑡−𝑢ℎ𝑡,𝜉𝑡||𝑑𝜏≤𝐶𝑡0‖𝛼‖2+‖𝛽‖2+‖‖𝛼𝑡‖‖2+‖‖𝛽𝑡‖‖2+‖‖𝜉𝑡‖‖2||||𝑑𝜏,𝑡0𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜃𝑡,𝜉𝑡||||=||||𝑑𝜏𝑡0𝑎𝑢𝑢ℎ𝑢ℎ𝑡−𝑢𝑡𝜃𝑡,𝜉𝑡𝑑𝜏+𝑡0𝑎𝑢𝑢ℎ𝑢𝑡𝜃𝑡,𝜉𝑡||||‖‖𝜉𝑑𝜏≤𝐶𝑡‖‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝛼𝑡‖‖2+‖‖𝛽𝑡‖‖2+‖‖𝜃𝑡‖‖2𝑑𝜏+𝐶𝑡0‖‖𝜃𝑡‖‖2+‖‖𝜉𝑡‖‖2||||𝑑𝜏,𝑡0𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜃,𝜉𝑡||||≤𝑑𝜏𝑡0||𝑏𝑢𝑢ℎ𝑢ℎ𝑡−𝑢𝑡𝜃,𝜉𝑡||𝑑𝜏+𝑡0||𝑏𝑢𝑢ℎ𝑢𝑡𝜃,𝜉𝑡||‖‖𝜉𝑑𝜏≤𝐶𝑡‖‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝛼𝑡‖‖2+‖‖𝛽𝑡‖‖2+‖𝜃‖2𝑑𝜏+𝐶𝑡0‖𝜃‖2+‖‖𝜉𝑡‖‖2𝑑𝜏.(3.15)Inserting (3.12)–(3.15) into (3.11) and using the Cauchy-Schwartz inequality lead to
𝑡0‖∇⋅𝜁‖21𝑑𝜏+2𝑎𝑢ℎ𝜉𝑡,𝜉𝑡+𝑏𝑢ℎ𝜉,𝜉𝑡≤𝐶‖𝜂‖2+‖‖𝜉𝑡‖‖2+‖𝛼‖2+‖𝛽‖2+‖𝜃‖2+‖‖𝜃𝑡‖‖2+𝐶𝑡0‖‖𝜃𝑡𝑡‖‖2+‖‖𝜃𝑡‖‖2+‖‖𝜉𝑡‖‖2+‖‖𝜂𝑡‖‖2+‖𝛼‖2+‖𝛽‖2+‖𝜁‖2‖‖𝜉𝑑𝜏+𝐶𝑡‖‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝛼𝑡‖‖2+‖‖𝛽𝑡‖‖2+‖‖𝜉𝑡‖‖2‖‖𝜉𝑑𝜏+𝐶𝑡‖‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖𝜉‖2𝑑𝜏+𝐶𝑡0‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖𝜉‖2+‖‖𝜉𝑡‖‖2𝑑𝜏.(3.16)
Integrating (3.16) from 0 to 𝑡, using the fact (𝑏(𝑢ℎ)𝜉,𝜉𝑡)=(1/2)(𝑑/𝑑𝑡)(𝑏(𝑢ℎ)𝜉,𝜉)−(1/2)(𝑏𝑢(𝑢ℎ)𝑢ℎ𝑡𝜉,𝜉𝑡) and the inequality
𝑡0𝜏0||||𝜓(𝑠)2𝑑𝑠𝑑𝜏≤𝐶𝑡0||||𝜓(𝑠)2𝑑𝑠,(3.17)
yields
‖𝜉‖2‖‖𝜉≤𝐶𝑡‖‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝛼𝑡‖‖2+‖‖𝛽𝑡‖‖2+‖‖𝜉𝑡‖‖2‖‖𝜉𝑑𝜏+𝐶𝑡‖‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖𝜉‖2𝑑𝜏+𝐶𝑡0‖‖𝛼𝑡‖‖2+‖‖𝛽𝑡‖‖2+‖𝛼‖2+‖𝛽‖2+‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖‖𝜃𝑡𝑡‖‖2+‖‖𝜂𝑡‖‖2+‖𝜉‖2+‖‖𝜉𝑡‖‖2+‖𝜁‖2𝑑𝜏.(3.18)
Thus, to estimate ‖𝜉‖, we need to estimate ‖𝛽‖, ‖𝛽𝑡‖, ‖𝜁‖, and ‖𝜉𝑡‖. Taking 𝑣ℎ=𝛽 in (3.5) leads to
(∇𝛽,∇𝛽)=(𝜉,∇𝛽)+(𝜃,∇𝛽).(3.19)
By the Cauchy-Schwartz inequality, we obtain
‖∇𝛽‖≤𝐶(‖𝜉‖+‖𝜃‖).(3.20)
Note that 𝛽∈𝑉ℎ⊂𝐻10(Ω) and ‖𝛽‖≤𝐶‖∇𝛽‖. We further have
‖𝛽‖≤𝐶(‖𝜉‖+‖𝜃‖).(3.21)
Differentiating (3.5) with respect to 𝑡 and choosing 𝑣ℎ=𝛽𝑡 gives
‖‖∇𝛽𝑡‖‖‖‖𝜉≤𝐶𝑡‖‖+‖‖𝜃𝑡‖‖.(3.22)
Similarly, since 𝛽∈𝑉ℎ⊂𝐻10(Ω), one has ‖𝛽𝑡‖≤‖∇𝛽𝑡‖≤𝐶(‖𝜉𝑡‖+‖𝜃𝑡‖).Taking 𝐰ℎ=𝜁 in (3.6), one has
𝑎𝑢(𝜁,𝜁)=ℎ𝜉𝑡+𝑏𝑢,𝜁ℎ+𝝈𝜉,𝜁𝑡𝑎𝑢(𝑢)−𝑎ℎ+𝝈𝑢,𝜁𝑏(𝑢)−𝑏ℎ+𝑎𝑢,𝜁ℎ𝜃𝑡+𝑏𝑢,𝜁ℎ𝜃,𝜁−(𝜂,𝜁).(3.23)
By the Cauchy-Schwartz inequality, we obtain
‖‖𝜃‖𝜁‖≤𝐶‖𝜉‖+𝑡‖‖‖‖𝜉+‖𝜃‖+‖𝜂‖+‖𝛼‖+‖𝛽‖+𝑡‖‖.(3.24)
To bound ‖𝜉𝑡‖2, we differentiate (3.6) with respect to 𝑡 to obtain
𝜁𝑡,𝐰ℎ−𝑎𝑢ℎ𝜉𝑡𝑡,𝐰ℎ−𝑏𝑢ℎ𝜉𝑡,𝐰ℎ=𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜉𝑡,𝐰ℎ+𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜉,𝐰ℎ+𝝈𝑡𝑡𝑎𝑢(𝑢)−𝑎ℎ,𝐰ℎ+𝝈𝑡𝑎𝑢(𝑢)𝑢𝑡−𝑎𝑢𝑢ℎ𝑢ℎ𝑡,𝐰ℎ+𝝈𝑡𝑢𝑏(𝑢)−𝑏ℎ,𝐰ℎ+𝝈𝑏𝑢(𝑢)𝑢𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡,𝐰ℎ+𝑎𝑢ℎ𝜃𝑡𝑡,𝐰ℎ+𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜃𝑡,𝐰ℎ+𝑏𝑢ℎ𝜃𝑡,𝐰ℎ+𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜃,𝐰ℎ−𝜂𝑡,𝐰ℎ,∀𝐰ℎ∈𝐻ℎ.(3.25)
Testing (3.25) with 𝐰ℎ=𝜉𝑡𝑡 and (3.4) with 𝐪ℎ=𝜁𝑡 and combining the resulting equations together lead to
∇⋅𝜁,∇⋅𝜁𝑡+𝑎𝑢ℎ𝜉𝑡𝑡,𝜉𝑡𝑡+𝑏𝑢ℎ𝜉𝑡,𝜉𝑡𝑡𝑎=−𝑢𝑢ℎ𝑢ℎ𝑡𝜉𝑡,𝜉𝑡𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜉,𝜉𝑡𝑡−𝝈𝑡𝑡𝑎𝑢(𝑢)−𝑎ℎ,𝜉𝑡𝑡−𝝈𝑡𝑎𝑢(𝑢)𝑢𝑡−𝑎𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡𝑡−𝝈𝑡𝑢𝑏(𝑢)−𝑏ℎ,𝜉𝑡𝑡−𝝈𝑏𝑢(𝑢)𝑢𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡𝑡−𝑎𝑢ℎ𝜃𝑡𝑡,𝜉𝑡𝑡−𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜃𝑡,𝜉𝑡𝑡−𝜃𝑡,𝜁𝑡−𝑏𝑢ℎ𝜃𝑡,𝜉𝑡𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜃,𝜉𝑡𝑡+𝜂𝑡,𝜉𝑡𝑡.(3.26)
Note that
𝑏𝑢ℎ𝜉𝑡,𝜉𝑡𝑡=12𝑑𝑏𝑢𝑑𝑡ℎ𝜉𝑡,𝜉𝑡−12𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜉,𝜉𝑡,∇⋅𝜁,∇⋅𝜁𝑡=12𝑑𝜃𝑑𝑡(∇⋅𝜁,∇⋅𝜁),𝑡,𝜁𝑡=𝑑𝜃𝑑𝑡𝑡−𝜃,𝜁𝑡𝑡.,𝜁(3.27)
Thus, (3.26) can be rewritten as
12𝑑𝑎𝑢𝑑𝑡(∇⋅𝜁,∇⋅𝜁)+ℎ𝜉𝑡𝑡,𝜉𝑡𝑡+12𝑑𝑏𝑢𝑑𝑡ℎ𝜉𝑡,𝜉𝑡𝑎=−𝑢𝑢ℎ𝑢ℎ𝑡𝜉𝑡,𝜉𝑡𝑡−12𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜉,𝜉𝑡𝑡−𝝈𝑡𝑡𝑢𝑎(𝑢)−𝑎ℎ,𝜉𝑡𝑡−𝝈𝑡𝑎𝑢(𝑢)𝑢𝑡−𝑎𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡𝑡−𝝈𝑡𝑏𝑢(𝑢)−𝑏ℎ,𝜉𝑡𝑡−𝝈𝑏𝑢(𝑢)𝑢𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡𝑡−𝑎𝑢ℎ𝜃𝑡𝑡,𝜉𝑡𝑡−𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜃𝑡,𝜉𝑡𝑡−𝑑𝜃𝑑𝑡𝑡+𝜃,𝜁𝑡𝑡−𝑏𝑢,𝜁ℎ𝜃𝑡,𝜉𝑡𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜃,𝜉𝑡𝑡+𝜂𝑡,𝜉𝑡𝑡.(3.28)
Integrating (3.28) from 0 to 𝑡 yields
(∇⋅𝜁,∇⋅𝜁)+𝑡0𝑎𝑢ℎ𝜉𝑡𝑡,𝜉𝑡𝑡+𝑏𝑢ℎ𝜉𝑡,𝜉𝑡=−𝑡0𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜉𝑡,𝜉𝑡𝑡1𝑑𝜏−2𝑡0𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜉,𝜉𝑡𝑡−𝑑𝜏𝑡0𝝈𝑡𝑡𝑢𝑎(𝑢)−𝑎ℎ,𝜉𝑡𝑡𝑑𝜏−𝑡0𝝈𝑡𝑎𝑢(𝑢)𝑢𝑡−𝑎𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡𝑡−𝑑𝜏𝑡0𝝈𝑡𝑢𝑏(𝑢)−𝑏ℎ,𝜉𝑡𝑡𝑑𝜏−𝑡0𝝈𝑏𝑢(𝑢)𝑢𝑡−𝑏𝑢𝑢ℎ𝑢ℎ𝑡,𝜉𝑡𝑡−𝑑𝜏𝑡0𝑎𝑢ℎ𝜃𝑡𝑡,𝜉𝑡𝑡𝑑𝜏−𝑡0𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜃𝑡,𝜉𝑡𝑡𝜃𝑑𝜏−𝑡+,𝜁𝑡0𝜃𝑡𝑡−,𝜁𝑑𝜏𝑡0𝑏𝑢ℎ𝜃𝑡,𝜉𝑡𝑡𝑑𝜏−𝑡0𝑏𝑢𝑢ℎ𝑢ℎ𝑡𝜃,𝜉𝑡𝑡𝑑𝜏+𝑡0𝜂𝑡,𝜉𝑡𝑡𝑑𝜏.(3.29)For the first term on the right-hand side of (3.29), by the Cauchy-Schwarz inequality and Young's inequality, for sufficiently small constant 𝜀>0, it holds that
||||−𝑡0𝑎𝑢𝑢ℎ𝑢ℎ𝑡𝜉𝑡,𝜉𝑡𝑡||||≤||||𝑑𝜏𝑡0𝑎𝑢𝑢ℎ𝑢ℎ𝑡−𝑢𝑡𝜉𝑡,𝜉𝑡𝑡||||+||||𝑑𝜏𝑡0𝑎𝑢𝑢ℎ𝑢𝑡𝜉𝑡,𝜉𝑡𝑡||||‖‖𝜉𝑑𝜏≤𝐶𝑡‖‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝛼𝑡‖‖2+‖‖𝛽𝑡‖‖2𝑑𝜏+𝐶𝑡0‖‖𝜉𝑡‖‖2‖‖𝜉𝑑𝜏+𝜀1+𝑡‖‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝜉𝑡𝑡‖‖2𝑑𝜏.(3.30)
Similarly, we can bound (3.29) as follows:
‖∇⋅𝜁‖2+‖‖𝜉𝑡‖‖2+𝑡0‖‖𝜉𝑡𝑡‖‖2‖‖𝜉𝑑𝜏≤𝐶𝑡‖‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝛼𝑡‖‖2+‖‖𝛽𝑡‖‖2‖‖𝜉𝑑𝜏+𝜀1+𝑡‖‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝜉𝑡𝑡‖‖2𝑑𝜏+𝐶‖𝜉‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝛼𝑡‖‖2+‖‖𝛽𝑡‖‖2𝑑𝜏+𝜀1+‖𝜉‖𝐿∞(0,𝑡;𝐿∞)𝑡0‖‖𝜉𝑡𝑡‖‖2𝑑𝜏+𝐶𝑡0‖𝜉‖2+‖𝜁‖2+‖‖𝜃𝑡𝑡‖‖2+‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖𝛼‖2+‖‖𝛼𝑡‖‖2+‖𝛽‖2+‖‖𝛽𝑡‖‖2+‖‖𝜂𝑡‖‖2+‖‖𝜉𝑡‖‖2+‖‖𝜃𝑑𝜏𝑡‖‖‖𝜁‖.(3.31)
In the following error analysis, we make an induction hypothesis:
‖‖𝜉𝑡‖‖𝐿∞(0,𝑡;𝐿∞)+‖𝜉‖𝐿∞(0,𝑡;𝐿∞)≤1.(3.32)
Utilizing (3.32), (3.24), (3.22), (3.21), and Young's inequality, one can reduce (3.31) to
‖∇⋅𝜁‖2+‖‖𝜉𝑡‖‖2‖≤𝐶𝜉‖2+‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖𝛼‖2+‖𝜂‖2+𝐶𝑡0‖𝜉‖2+‖‖𝜃𝑡𝑡‖‖2+‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖𝛼‖2+‖‖𝛼t‖‖2+‖‖𝜂𝑡‖‖2+‖‖𝜉𝑡‖‖2𝑑𝜏.(3.33)
Then by Gronwall's inequality, we obtain
‖∇⋅𝜁‖2+‖‖𝜉𝑡‖‖2‖≤𝐶𝜉‖2+‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖𝛼‖2+‖𝜂‖2+𝐶𝑡0‖𝜉‖2+‖‖𝜃𝑡𝑡‖‖2+‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖𝛼‖2+‖‖𝛼𝑡‖‖2+‖‖𝜂𝑡‖‖2𝑑𝜏.(3.34)
Furthermore, by (3.24) and (3.34), one has
‖𝜁‖2‖≤𝐶𝜉‖2+‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖𝛼‖2+‖𝜂‖2+𝐶𝑡0‖𝜉‖2+‖‖𝜃𝑡𝑡‖‖2+‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖𝛼‖2+‖‖𝛼𝑡‖‖2+‖‖𝜂𝑡‖‖2𝑑𝜏.(3.35)
Therefore, by the estimates of ‖𝛽‖, ‖𝛽𝑡‖, ‖𝜁‖, and ‖𝜉𝑡‖, it follows that
‖𝜉‖2≤𝐶𝑡0‖𝜉‖2+‖‖𝜃𝑡𝑡‖‖2+‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖𝛼‖2+‖‖𝛼𝑡‖‖2+‖𝜂‖2+‖‖𝜂𝑡‖‖2𝑑𝜏.(3.36)
Applying Gronwall's inequality to the above equation and using the estimates of projection operators give
‖𝜉‖2≤𝐶𝑡0‖‖𝜃𝑡𝑡‖‖2+‖‖𝜃𝑡‖‖2+‖𝜃‖2+‖𝛼‖2+‖‖𝛼𝑡‖‖2+‖𝜂‖2+‖‖𝜂𝑡‖‖2𝑑𝜏≤𝐶ℎmin(2𝑘+2,2𝑚+2)‖‖𝑢𝑡‖‖2𝐿∞𝐻𝑚+1+‖𝑢‖2𝐿∞𝐻𝑚+1+‖‖𝐩𝑡‖‖2𝐿∞𝐻𝑘+1+‖𝐩‖2𝐿∞(𝐻𝑘+1)+‖‖𝝈𝑡‖‖2𝐿∞(𝐻𝑘+1)+‖‖𝝈𝑡𝑡‖‖2𝐿∞(𝐻𝑘+1)+‖𝝈‖2𝐿∞𝐻𝑘+1.(3.37)
Inserting the estimate of ‖𝜉‖ into (3.34) yields
‖‖𝜉𝑡‖‖2≤𝐶ℎmin(2𝑘+2,2𝑚+2).(3.38)
Thus, the estimates of 𝛽 and 𝜁 follow from the estimate of 𝜉.Finally, according to the proof of the induction hypothesis in [23, 30], we can prove that the inductive hypothesis (3.32) holds. In fact, when 𝑡=0, then 𝜉(0)=0, 𝜉𝑡(0)=0. Note that ‖𝜉‖𝐿∞(0,𝑡;𝐿∞)+‖𝜉𝑡‖𝐿∞(0,𝑡;𝐿∞) is continuous w.r.t. 𝑡. Then, we conclude that there exists 𝑡1∈(0,𝑇] such that
‖𝜉‖𝐿∞(0,𝑡1;𝐿∞)+‖‖𝜉𝑡‖‖𝐿∞(0,𝑡1;𝐿∞)≤1.(3.39)
Set 𝑡∗=sup𝑡1. Thus, ‖𝜉‖𝐿∞(0,𝑡∗;𝐿∞)+‖𝜉𝑡‖𝐿∞(0,𝑡∗;𝐿∞)≤1. Therefore, we have
‖‖𝜉𝑡∗‖‖+‖‖𝜉𝑡𝑡∗‖‖≤𝐶ℎmin(𝑘+1,𝑚+1).(3.40)By inverse estimates, we deduce that, for any 0≤𝑡≤𝑡∗, it holds that
‖𝜉‖𝐿∞(0,𝑡;𝐿∞)+‖‖𝜉𝑡‖‖𝐿∞(0,𝑡;𝐿∞)≤𝐶ℎmin(𝑘+1,𝑚+1)−𝑑/2.(3.41)
Then we can take ℎ>0 sufficiently small such that
‖𝜉‖𝐿∞(0,𝑡∗;𝐿∞)+‖‖𝜉𝑡‖‖𝐿∞(0,𝑡∗;𝐿∞)<1.(3.42)
Again, by the continuity of ‖𝜉‖𝐿∞(0,𝑡;𝐿∞)+‖𝜉𝑡‖𝐿∞(0,𝑡;𝐿∞), we conclude that there exists a positive constant 𝛿 such that
‖𝜉‖𝐿∞(0,𝑡∗+𝛿;𝐿∞)+‖‖𝜉𝑡‖‖𝐿∞(0,𝑡∗+𝛿;𝐿∞)≤1,(3.43)
which contracts to the definition of 𝑡∗. This completes the proof of the induction hypothesis.Combining (3.21), (3.37), (3.2), (2.26), (2.27) with the estimates of auxiliary projections and utilizing the triangle inequality, we can derive the desired result.

Remark 3.2. By Theorem 3.1 and the standard embedding theorem, we can obtain the 𝐿∞ estimate for 𝑑=1 and 2 as follows:
‖‖𝑢−𝑢ℎ‖‖𝐿∞(𝐿∞)≤𝐶2||||lnℎ𝑑−1ℎmin(𝑘+1,𝑚+1).(3.44)

4. Conclusion

In this paper, 𝐻1-Galerkin mixed finite element method combining with expanded mixed element method is discussed for nonlinear viscoelasticity equations. This method solves the scalar unknown, its gradient, and its flux, directly. It is suitable for the case that the coefficient of the differential is a small tensor and does not need to be inverted. Furthermore, the formulation permits the use of standard continuous and piecewise (linear and higher-order) polynomials in contrast to continuously differentiable piecewise polynomials required by the standard 𝐻1-Galerkin methods and is free of the LBB condition which is required by the mixed finite element methods.

There are also some important issues to be addressed in the area; for example, one can consider numerical implementation and mathematical and numerical analysis of the full discrete procedure. This is an important and challenging topic in the future research.